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If 4x(5n) = t and x and n are integers, what is the value of t ?

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If 4x(5n) = t and x and n are integers, what is the value of t ? [#permalink]

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New post 21 Dec 2016, 05:33
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If 4x(5n) = t and x and n are integers, what is the value of t ?

(1) t is a perfect square less than 250.
(2) xn = 5
[Reveal] Spoiler: OA

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If 4x(5n) = t and x and n are integers, what is the value of t ? [#permalink]

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New post 21 Dec 2016, 05:43
Official Explanation:

Quote:
4x(5n) is equivalent to 20xn, so to fi nd the value of t, you need
the value of xn. Statement (1) is sufficient: there is only one perfect square that
is less than 250 and divisible by 20: 100. (t must be divisible by 20 because x
and n must be integers.) Statement (2) is also sufficient: it directly gives us
the information we need: the value of xn. Choice (D) is correct.


For (1): While i understand that 100 is the only perfect square less than 250 and divisible by 20, i would like to ask you guys what´s wrong with the following approach:

Given from (1): 20xy < 250 -> xy < 12.5

So there are basically two perfect squares below 12.5 -> 4 and 9. So x and y (both integers) are either 2 and 2 or 3 and 3, hence the statement should be insufficient.

Can anyone (...or @Bunuel) explain the mistake in my thought process?

Thank you!

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Re: If 4x(5n) = t and x and n are integers, what is the value of t ? [#permalink]

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New post 21 Dec 2016, 05:59
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paddy41 wrote:
If 4x(5n) = t and x and n are integers, what is the value of t ?

(1) t is a perfect square less than 250.
(2) xn = 5


4 is already a perfect square, so our \(x\) can be only of the form \(2^{even} = 2^0, 2^2, 2^4\) ...

5 is not a perfect square, so in order for \(5n\)to be a perfect square \(n\) should be of the form \(5^{odd} = 5, 5^3, 5^5\)...

(1) t is a perfect square less than 250

We can have following options:

(4*1)*(5*5) = 100 < 250

(4*4) * (5*5) = 400 > 250

(4*1)*(5*125) = 2500 > 250
...
All other options will be greater than 250. Sufficient.

(2) xn = 5

We have only one option which gives t as a pefect square - \(2^0*5^1\). Hence x=1, n=5

4*25 = 100. Sufficient.

Answer D

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If 4x(5n) = t and x and n are integers, what is the value of t ? [#permalink]

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New post 21 Dec 2016, 06:38
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BOOKMARKED
paddy41 wrote:
Official Explanation:

Quote:
4x(5n) is equivalent to 20xn, so to fi nd the value of t, you need
the value of xn. Statement (1) is sufficient: there is only one perfect square that
is less than 250 and divisible by 20: 100. (t must be divisible by 20 because x
and n must be integers.) Statement (2) is also sufficient: it directly gives us
the information we need: the value of xn. Choice (D) is correct.


For (1): While i understand that 100 is the only perfect square less than 250 and divisible by 20, i would like to ask you guys what´s wrong with the following approach:

Given from (1): 20xy < 250 -> xy < 12.5

So there are basically two perfect squares below 12.5 -> 4 and 9. So x and y (both integers) are either 2 and 2 or 3 and 3, hence the statement should be insufficient.


Can anyone (...or @Bunuel) explain the mistake in my thought process?

Thank you!


I don't think that inequality is the right way to solve this question.

I'd better put it this way:

\(t^2 = 20xn\)

\(t = \sqrt{20xn} = 2*\sqrt{5xn}\)

For the expression under the radical to be an integer min value of \(xn\) can be \(5\). Either \(x=1\) and \(n=5\) or \(x=5\) and \(n=1\). In both cases we'll get our \(t\).

If you wish you can put it in the form of inequality:

\(2*\sqrt{5xn} < \sqrt{250}\)

or

\(2*\sqrt{5xn} < 5* \sqrt{10}\)

But I don't think this will help much, because you'll need to approximate \(5* \sqrt{10}\).

Regards

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Re: If 4x(5n) = t and x and n are integers, what is the value of t ? [#permalink]

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New post 20 Apr 2017, 21:22
paddy41 wrote:
If 4x(5n) = t and x and n are integers, what is the value of t ?

(1) t is a perfect square less than 250.
(2) xn = 5


Hi,

Given \(t = 20xn, t=?\)

(1) t is a perfect square less than 250.

\(t = 20xn = 2^{2}*5*x*n\), In order to \(t\) to be a perfect square either \((x = 5, n = 1) \textrm{or} (n = 5, x = 1). => t = 100\)

Hence, Sufficient.

(2) xn = 5 => This statement is clearly sufficient.

Answer: (D)

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Re: If 4x(5n) = t and x and n are integers, what is the value of t ? [#permalink]

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New post 22 Apr 2017, 16:46
Given t = 20xn

Basic inferences
i. t is even
ii. units digit o t = 0

Statement 1
t is a perfect square less than 250.
100 is the only perfect square below 250 such that the units digit is 0
hence A is sufficient

Statement 2
XN = 5
we can calculate value of 20xn
hence B alone is sufficient

Final Answer: D

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Re: If 4x(5n) = t and x and n are integers, what is the value of t ? [#permalink]

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New post 21 Dec 2017, 10:27
Ans B.

St1: t=20xn is a perfect square below 250 and x&n are integers (can be positive, negative or zero)
Therefore, there are two possible perfect squared values for t: 0 (x*n=0) and 100 (x*n=5)
Insufficient

St.2: xn=5; Therefore, t=100. Sufficient

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If 4x(5n) = t and x and n are integers, what is the value of t ? [#permalink]

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New post 21 Dec 2017, 11:00
Shobhit7 wrote:
Ans B.

St1: t=20xn is a perfect square below 250 and x&n are integers (can be positive, negative or zero)
Therefore, there are two possible perfect squared values for t: 0 (x*n=0) and 100 (x*n=5)
Insufficient

St.2: xn=5; Therefore, t=100. Sufficient


Hi Shobhit7

Well you have put forward an interesting perspective here. Some consider 0 to be a perfect square and some not.

A perfect square is always positive. If 0 is a perfect square, then it will be positive. but we know as per GMAT 0 is neither positive nor negative.

Hi Bunuel

The question seems to be ambiguous and the solution can have multiple interpretations. Can you clarify what is GMAT's take on 0 as a perfect square?

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Joined: 02 Sep 2009
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Re: If 4x(5n) = t and x and n are integers, what is the value of t ? [#permalink]

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New post 21 Dec 2017, 20:09
niks18 wrote:
Shobhit7 wrote:
Ans B.

St1: t=20xn is a perfect square below 250 and x&n are integers (can be positive, negative or zero)
Therefore, there are two possible perfect squared values for t: 0 (x*n=0) and 100 (x*n=5)
Insufficient

St.2: xn=5; Therefore, t=100. Sufficient


Hi Shobhit7

Well you have put forward an interesting perspective here. Some consider 0 to be a perfect square and some not.

A perfect square is always positive. If 0 is a perfect square, then it will be positive. but we know as per GMAT 0 is neither positive nor negative.

Hi Bunuel

The question seems to be ambiguous and the solution can have multiple interpretations. Can you clarify what is GMAT's take on 0 as a perfect square?


0 is a perfect square. The question is flawed.
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Re: If 4x(5n) = t and x and n are integers, what is the value of t ?   [#permalink] 21 Dec 2017, 20:09
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